The most obvious point is that there would be very little darkness on the surface of the world, at best you would get twilight twice a "day" unless there was a large axial tilt that would give a true day/night cycle at the poles. If the stars were sufficiently alike, they could well be indistinguishable from one-another, or if they are different, there could be periods of alternating colour i.e. blue days and red days, and/or bright days and dim days.
The world would have to be a sufficient distance away that radiated heat would pretty much equal radiant influx, so the suns would appear rather smaller than is the case with Earth and Sol. This would mean that in sunny weather, shadows would be hard-edged except at twilight, when there would be hardly any shadows at all, just dim light. With any sort of atmospheric scattering of light, the stars would never be seen except at the poles if there was sufficient axial tilt, so the sky would most likely vary only between blue and bright and pink/red and dull. Since the stars would not be visible from most of the world, this would place a more inward-oriented bias to the beliefs of sentient inhabitants.
Caves, being the only place that true darkness would ever be seen in most regions, are likely to be a source of wonder and terror to sentient surface-dwellers, and cave dwellers would hardly ever venture out into the sunlit areas where they could be seen. Life-forms would probably not evolve cyclic sleep-wake patterns, as dim periods would be relatively brief, so if any creature needed to sleep, it would have to do so in a safe place, guarded by wakeful friends or family, or be able to defend itself despite being asleep. It is likely that sleep would be something more like hibernation, if it occurred at all.
This is, of course, ignoring the fact that the barycenter of a binary pair of stars is an unstable place for a planet to be. It would be all to easy for the planet to slide out of the barycenter and fall into one or other of the stars, or at the very least end up in a close orbit around one where things would quickly get much, much hotter... At least, the world itself would not be able to have a moon, given the additional instability this would cause. However, the forces required to keep a world within the barycenter of a binary star system would be relatively small, so it is not impossible if there is some form of intervention, presumably of sentient origin.
Since light would be more-or-less omnipresent, I disagree with Beta Decays now-deleted answer that there would be one religion worshipping each star, unless the stars were very different in colour and/or size. A more likely focus of religion would be the darkness within caves (or the light outside them) - light-dwellers would be likely to have practically no night-vision at all, and cave dwellers would most likely not have very good sight and would be vulnerable to light-dwelling predators with good vision. From either direction, the opposite environment would be a mystery.